1. Edit TTL's on DNS and wait 8 hours.
2. Use cPanel's multisite copy tool to transfer sites. Depending on quantity you might want to do it in batches.
3. Any failed transfers do by hand via SSH using cPanel's backup/restore scripts.
4. Edit DNS zones to reflect new IP's.
5. Change DNS settings with registrar.
6. Wait 3 days and shut off the old server.
If you need any assistance doing this, please let me know.